Student-mathematics versus teacher-Metamatics
Tarp, Allan, action researcher, Grenaa, Denmark, , version 1.2
Paper presented at the ethnography network’s symposium on learners inclusion at the ECER conference September 2003,
Key terms: mathematics, postmodern, calculus, learner inclusion, ethnography, action research, counter-research, teacher education, e-learning
Abstract
The writer reports on his career as an action researcher helping the students to develop their own student-mathematics making mathematics accessible for all but being opposed by the educational system. The work took place over a 30 year-period in Danish calculus and pre-calculus classes and in Danish teacher education. As methodology a postmodern counter-research was developed accepting number-statements but being sceptical towards word-statements. Counter-research sees word-researchers as counsellors in a courtroom of correctness. The modern researcher is a counsellor for the prosecution trying to produce certainty by accusing things of being something, and the postmodern researcher is a counsellor for the defence trying to produce doubt by listening to witnesses, and by cross-examining to look for hidden differences that might make a difference. A micro-curriculum in student mathematics was developed and tested in 13 grade 11 classes showing a high degree of improvement in student performance.
A Confession
I confess I have always listened to the students. ‘Ich bin ein Berliner’ John F. Kennedy said. We are all students, and maybe we should stay sceptical students and keep on learning; and wait to teach until we have found something that is certain, and cannot be different[1].
I was studying mathematics at the university, but during my study I learned that there was another hidden mathematics different from the one in the textbook.
According to the Danish textbooks mathematics is something above the physical world, a metaphysical subject that is studied for its own sake to obtain pure knowledge; a subject that might, but has no need to, be applied to the real world.
But looking at Anglo-Saxon textbooks and at the history of mathematics I found out that the world is not applying mathematics, the world is creating mathematics. Mathematics was born as quantitative stories about multiplicity, just as the names Geometry and Algebra clearly indicates: ‘geometry’ means ‘earth-measuring’ in Greek, and ‘algebra’ means ‘reuniting’ in Arabic. Thus geometry and algebra are answers to the two fundamental questions that arose when humans went from gathering & hunting to agriculture: How do we divide our land, and our products.
This however was not how Geometry and Algebra was presented in the Danish textbooks. Here they were presented as examples of sets. Numbers were sets, calculations were sets, and all of mathematics was examples of sets.
To name this difference between textbook-mathematics and real-world mathematics I coined the word ‘meta-matics’ from above as opposed to ‘mathe-matics’ from below.
Until then I had seen no meaning in mathematics, which I had to learn more or less by heart, and I was planning to shift away from mathematics to study architecture instead. The discovery of mathematics from below however changed this. All of a sudden I found mathematics to be a fascinating number-language that could be applied to describe the world in numbers, which can be calculated and thus predicted. And I decided to share this excitement with others, thus choosing to continue my study and become a mathematics teacher.
And the summer before starting I wrote an alternative textbook in mathematics, Calculus as mathematics from below, as opposed to the traditional textbook, Calculus as meta-matics from above.
In my Calculus textbook I showed how mathematics grows out of real-world problems. I had expected my students to be as excited as I was. Instead they said: Are we going to learn about mathematics, or are we going to learn about real-world problems? Both, I answered. But what if the real-world problems can be solved in another way, they asked.
Now I was caught in a dilemma. The students made me realise that I was trying to sell meta-matics hidden under a thin surface of applications. Thus forcing them to learn two things, meta-matics and applications.
So I had to reject my Calculus-textbook, and do as the others, follow the norm. However most students did not understand the traditional textbook. So what should I do, should I turn to architecture, or should I cross over to help the students develop their own mathematics?
I confess I became a renegade. And for the next 7 years I continually wrote new texts adjusting mathematics to the students’ suggestions. Then finally we had found a text that worked so that all students were able to understand and learn calculus. I transformed this student-mathematics into a textbook, which I published, expecting that the ministry and the other teachers would welcome it as a solution to the low success rate in calculus.
But the other teachers neglected it; and the ministry told me that a textbook on calculus should cover 200 pages, and if I continued to use my 48 pages textbook I would be discharged.
So I had to reject this textbook on student-mathematics in calculus.
Instead I turned to pre-calculus, which has even bigger problems than calculus being a compulsory subject that most student find difficult to understand. Again I worked as an action researcher listening to the students’ suggestions. And again the traditional 200-page textbook in meta-matics was replaced by a minor textbook in student-mathematics covering 12 pages.
But this time I did not publish the textbook. Instead I applied for a PhD scholarship in order to try out part of it with other teachers in their classes. I asked for volunteer teachers to try out a 20 lesson introductory course in student-mathematics. And I was in luck; out of approximately 1500 mathematics teacher 1.5 volunteered, a full time teacher and a temporary teacher. So other teachers I had to persuade.
For three years I followed the two volunteer teachers teaching three classes each. The teachers had big problems leaving the tradition to practise the student-mathematics. Still student-mathematics turned out to be so robust that almost all students expressed satisfaction, some even surprise to be allowed to enter the field of mathematics, which had always been closed to them[2].
The full time teacher wanted to extend the student-mathematics to a full year programme, but the ministry turned his application down even if the ministry had called for experiments in order to save the pre-calculus mathematics, which was at risk to be terminated because around 50% of the students fail the written exam.
So I have returned to my own classroom to try out the full version of the student-mathematics myself, but again the ministry turned the application down so I had to do a little of both in the classroom. However this compromise proved to be only temporary since the external examiner complained to the ministry, that I was not following an ordinary textbook as the rest was doing. And the ministry will probably echo its standard answer ‘follow the norm, or go to the dorm’.
So I also had to reject this textbook on student-mathematics in pre-calculus.
However there is a big advantage working in the research field. At the yearly teacher conference nobody wants to listen to people not following the norm[3]. At a research conference it is different. You don’t have to wait for an invitation that never comes; you can submit your own paper. After a presentation a researcher approached me. He had heard my presentation at two conferences and he was astonished that it made so much sense even if it was outside the traditional discourse on constructivist mathematics. He would like me to present student-mathematics to his students at his teacher college. So for one week we worked together presenting and translating student-mathematics to East-European students. They also were fascinated recommending that both traditional modern mathematics and student-mathematics (or postmodern mathematics as it was called) should be taught in teacher colleges[4].
And to my luck I was asked to teach a two-year e-learning course at a Danish teacher college. So here I had the opportunity to develop a full program in student-mathematics for teacher education.
The programme was successful with the students. But halfway through the programme my temporary job was transformed into a permanent job, which I could not get since the committee called me a missionary refusing to follow the norm. And instead of finishing the programme, the new teacher ordered a cure for this programme by asking the students to read the first 60 pages of the traditional textbook for their first meeting.
So I also had to reject this textbook on student-mathematics for teacher education.
Apparently there are big problems practising student-mathematics as long as meta-matics is in power. Just like the early mammals had to survive underground when the dinosaurs ruled the world. However I am in luck. The dinosaurs are about to make themselves extinct since they cannot reproduce. Mathematics education faces an enrolment crisis all over the world since only a few students want to study for mathematics-based educations, and even fewer want to become mathematics teachers[5].
So as the bird Phoenix raises again from the fire, I plan to make student-mathematics rise again as a virtual textbook to be placed on the Internet as a self-reproducing virus.
During my short life as at the teacher college I learned, that students studying student-mathematics do not need a teacher. Meta-matics from above needs a teacher as a transmitter since it places its authority in the metaphysical world above, from which meta-matics is supposed to flow through researchers and teachers to the students.
The student-mathematics places its authority in the physical world below, in multiplicity. Multiplicity can be studied in your own living room. All the teacher needs to do is to set up an agenda for an educational meeting between the student and the multiplicity. Then learning automatically takes places, both as tacit knowledge, competence, through a ‘sentence-free meeting with the sentence-subject’, and as discursive knowledge, qualifications, through a ‘sentence-loaded meeting with the sentence-subject’.
In this way 1 teacher can organise 16 students in 4 groups of 4 students acting by turns as instructors working together in pairs instructing the others, and being coached by one teacher over the internet.
Through pyramid-education each teacher continually produces 16 new teachers in student-mathematics, who pay for their education by each teaching a new group of 16 students. In this way student-mathematics will multiply on the Internet, until it can surface to the real world when the dinosaurs of meta-matics have died out from lack of fertility.
Now my confession ends. I am sorry that I left my tribe to join the others, those who are accused of being uneducated, uninterested, unruly, lazy, stupid, narcissistic, self-focused etc. etc. etc. to help them develop their own mathematics. But maybe this student-mathematics will survive once the 5000 years old subject mathematics has terminated its 100 years sidetrack of set-based meta-matics[6] and returned to multiplicity-based mathematics.
Methodology
The methodology of this action research grows out of institutional scepticism, as it appeared in the enlightenment and was implemented in its two democracies, the American in the form of pragmatism and symbolic interactionism, and the French in the form of post-structuralism and post-modernism[7]. This paper follows the postmodern scepticism towards logocentricity, i.e. towards the belief that the words represent the world[8].
I have developed a methodology called ‘postmodern counter-research’ based upon a post-structuralist ‘pencil-dilemma’: Placed between a ruler and a dictionary a thing can show its length, but not its name - hence a thing can falsify a number-statement about its length, but not a word-statement about its kind. I.e. a thing can defend itself against a number-accusation by making a statement of difference; but against a word-accusation it can only make a statement of deference. Unless it can ask for a counsellor for the defence, a postmodern counter-researcher.
A number is an ill written icon showing the degree of multiplicity (there are 4 strokes in the number sign 4, etc.); a word is a sound made by a person and recognised in some groups and not in others. Words can be questioned and put to a vote, numbers cannot. Numbers can carry valid conclusions based upon reliable data, i.e. research. Words can carry only interpretations, that if presented as research become seduction; words can not carry truth, only hide differences to be uncovered by postmodern counter research, having quality if the difference is a genuine ‘cinderella-difference’, i.e. a difference that makes a difference. Thus postmodern counter-research follows in the footsteps of the ancient sophists always distinguishing between what could be different and what could not.
This difference between numbers and words is socially recognised in the two social decision institutions, the laboratory and the courtroom. A number-statement is send to the laboratory to be decided upon by asking the thing through a measurement, and the judgement of the laboratory is final and cannot be appealed. A word-statement is send to a courtroom to be decided upon by the majority of votes in a jury; but the judgement of a courtroom is not final and can always be appealed, either to a higher courtroom or to the parliament asking it to change the law.
Thus word-researchers are not researchers but counsellors in a courtroom of correctness. Modern word-researchers are counsellors for the prosecution accusing the defendant of being guilty of being something, e.g. a pencil, or unable to learn mathematics. And trying to produce certainty about its ‘IS-claims’. Postmodern word-researchers are counter-researchers believing that no case can be proven. Hence counter-researchers always work as counsellors for the defence[9] listening to the defendant through narratives, and cross-examining the witnesses of the prosecution through interviews to find a deference hiding a difference. The aim is to produce so much doubt, that the benefit of the doubt should make the defendant acquitted; e.g. by finding a hidden difference that can be shown to make a difference.
The Case: Evidence and Cross-examination
In this case the students are being accused of being unable to learn pre-calculus mathematics[10]. To prove its case the prosecution has presented the mathematics textbook, that the students cannot reproduce at the oral exam, and statistics showing that almost half of the students fail the written exam. In the pledge the prosecution asks that mathematics should be x-rated to students over 16, unless the students are able to demonstrate special talents.
As a postmodern counter-researcher I act as a counsellor for the defence. As my first witness I call a Danish high school graduate Barbie. Barbie is asked to tell about her mathematics education in her own words:
In grade seven we were making graphs with negative and positive scales, how to draw them, and so when we asked why we made them, what purpose it kind of had, well you just had to make them, that’s how it was. You didn’t get any explanation as to the reality behind this mathematics. Our number two teacher, we had two different teachers that year, came in and was drunk as a lord. So we didn’t learn very much.
In the high school, where I had mathematics the first year, and I must say this was just what suited my head, at any case the teaching method was different, one I think should be spread out, for the teacher had a quite different way to explain, one you could understand. You really felt you learned something, even if it was difficult for you, you still learned it along the road. Even if you were a little behind, because first of all, you had a good relationship to the teacher, you felt the teacher was part of the class, not a separate part of the class thinking he has a higher authority. We really felt, the teacher was on the same level, as to authority any way, of course as to mathematics he was at a higher level. I do not know what I can explain about that method, anyway there was something about it that was incredibly attractive.
I can compare with mathematics the second year. The method, the teacher used the second year is simply one I find unsuitable and I know that many from the class agree. You felt precisely the opposite, the teacher was not so to speak a part of the class, you felt he was very authoritarian, he used his authority and taught directly from the book, and that helped us very little. When you go home and read the book and prepare your homework and then go back to school and say, that you don’t understand it, the teacher explains it and mostly it helps only little for he explains it directly as it is in the book. He could have turned it, but he didn’t.
Barbie describes different types of teachers. The last teacher ‘ taught directly from the book … he explains it directly as it is in the book. He could have turned it, but he didn’t.’ From this statement I will coin the words ‘echo-teaching’ and ‘the Math-bible’: The teacher enters the classroom, opens the textbook and begins to teach, but what he says is what is in the book. When he is asked to explain the book, he just repeats the book, thus practising ‘echo-teaching’. And by just repeating the book even when asked to explain it, the 'teacher shows that there is only one textbook, the Textbook, the Bible of mathematics, the ‘Math-bible’.